tensor(deprecated)/exterior_diff - Maple Help

tensor

 exterior_diff
 Compute the exterior derivative of a completely anti-symmetric covariant tensor.

 Calling Sequence exterior_diff(T, coord)

Parameters

 T - covariant anti-symmetric tensor or scalar coord - list of coordinate variable names

Description

Important: The tensor package has been deprecated. Use the superseding commands DifferentialGeometry[ExteriorDerivative] and Physics[ExteriorDerivative] instead.

 • The function exterior_diff( T, coord ) computes the exterior derivative of the (covariant) components of the anti-symmetric tensor T and returns them as a tensor_type of rank equal to $\mathrm{rank}\left(T\right)+1$.  The result is totally anti-symmetric and uses the antisymmetric indexing function (unless T is a scalar).
 • If T is not a scalar or vector, it should be indexed using the antisymmetric indexing function.  If it is not indexed using the antisymmetric indexing function and it is not a scalar or vector, then the routine will determine its anti-symmetric part and compare that with the tensor to see if the components are really totally anti-symmetric. If it is found not to be completely anti-symmetric, the routine exits with an error.
 • The result is computed first by finding the first partials of the tensor components and then antisymmetrizing them.
 • Simplification:  This routine uses the tensor/lin_com/simp and tensor/partial_diff/simp routines for simplification purposes.  The simplification routines are used internally by the partial_diff and antisymmetrize routines as they are called by exterior_diff.  By default, tensor/lin_com/simp and tensor/partial_diff/simp are initialized to the tensor/simp routine.  It is recommended that these routines be customized to suit the needs of the particular problem.
 • This command is part of the tensor package, so it can be used in the form exterior_diff(..) only after executing the command with(tensor). However, it can always be accessed through the long from of the command by using tensor[exterior_diff](..).

Examples

Important: The tensor package has been deprecated. Use the superseding packages DifferentialGeometry and Physics instead.

 > $\mathrm{with}\left(\mathrm{tensor}\right):$

Define the coordinates and an arbitrary skew-symmetric second rank tensor:

 > $\mathrm{coord}≔\left[x,y,z\right]:$
 > $\mathrm{tc}≔\mathrm{array}\left(\mathrm{antisymmetric},1..3,1..3\right):$
 > $\mathbf{for}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}i\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{to}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}3\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathbf{for}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}j\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{from}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}i+1\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{to}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}3\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathrm{tc}\left[i,j\right]≔\mathrm{cat}\left('t',i,j\right)\left(x,y,z\right)\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do};$$T≔\mathrm{create}\left(\left[-1,-1\right],\mathrm{eval}\left(\mathrm{tc}\right)\right)$
 ${T}{≔}{table}{}\left(\left[{\mathrm{index_char}}{=}\left[{-1}{,}{-1}\right]{,}{\mathrm{compts}}{=}\left[\begin{array}{ccc}{0}& {\mathrm{t12}}{}\left({x}{,}{y}{,}{z}\right)& {\mathrm{t13}}{}\left({x}{,}{y}{,}{z}\right)\\ {-}{\mathrm{t12}}{}\left({x}{,}{y}{,}{z}\right)& {0}& {\mathrm{t23}}{}\left({x}{,}{y}{,}{z}\right)\\ {-}{\mathrm{t13}}{}\left({x}{,}{y}{,}{z}\right)& {-}{\mathrm{t23}}{}\left({x}{,}{y}{,}{z}\right)& {0}\end{array}\right]\right]\right)$ (1)

Now compute the exterior derivative:

 > $\mathrm{ex_der}≔\mathrm{exterior_diff}\left(T,\mathrm{coord}\right)$
 ${\mathrm{ex_der}}{≔}{table}{}\left(\left[{\mathrm{index_char}}{=}\left[{-1}{,}{-1}{,}{-1}\right]{,}{\mathrm{compts}}{=}{array}{}\left({\mathrm{antisymmetric}}{,}{1}{..}{3}{,}{1}{..}{3}{,}{1}{..}{3}{,}\left[\left({1}{,}{1}{,}{1}\right){=}{0}{,}\left({1}{,}{1}{,}{2}\right){=}{0}{,}\left({1}{,}{1}{,}{3}\right){=}{0}{,}\left({1}{,}{2}{,}{1}\right){=}{0}{,}\left({1}{,}{2}{,}{2}\right){=}{0}{,}\left({1}{,}{2}{,}{3}\right){=}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t12}}{}\left({x}{,}{y}{,}{z}\right){-}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t13}}{}\left({x}{,}{y}{,}{z}\right){+}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t23}}{}\left({x}{,}{y}{,}{z}\right){,}\left({1}{,}{3}{,}{1}\right){=}{0}{,}\left({1}{,}{3}{,}{2}\right){=}{-}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t12}}{}\left({x}{,}{y}{,}{z}\right){+}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t13}}{}\left({x}{,}{y}{,}{z}\right){-}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t23}}{}\left({x}{,}{y}{,}{z}\right){,}\left({1}{,}{3}{,}{3}\right){=}{0}{,}\left({2}{,}{1}{,}{1}\right){=}{0}{,}\left({2}{,}{1}{,}{2}\right){=}{0}{,}\left({2}{,}{1}{,}{3}\right){=}{-}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t12}}{}\left({x}{,}{y}{,}{z}\right){+}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t13}}{}\left({x}{,}{y}{,}{z}\right){-}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t23}}{}\left({x}{,}{y}{,}{z}\right){,}\left({2}{,}{2}{,}{1}\right){=}{0}{,}\left({2}{,}{2}{,}{2}\right){=}{0}{,}\left({2}{,}{2}{,}{3}\right){=}{0}{,}\left({2}{,}{3}{,}{1}\right){=}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t12}}{}\left({x}{,}{y}{,}{z}\right){-}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t13}}{}\left({x}{,}{y}{,}{z}\right){+}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t23}}{}\left({x}{,}{y}{,}{z}\right){,}\left({2}{,}{3}{,}{2}\right){=}{0}{,}\left({2}{,}{3}{,}{3}\right){=}{0}{,}\left({3}{,}{1}{,}{1}\right){=}{0}{,}\left({3}{,}{1}{,}{2}\right){=}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t12}}{}\left({x}{,}{y}{,}{z}\right){-}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t13}}{}\left({x}{,}{y}{,}{z}\right){+}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t23}}{}\left({x}{,}{y}{,}{z}\right){,}\left({3}{,}{1}{,}{3}\right){=}{0}{,}\left({3}{,}{2}{,}{1}\right){=}{-}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t12}}{}\left({x}{,}{y}{,}{z}\right){+}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t13}}{}\left({x}{,}{y}{,}{z}\right){-}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{t23}}{}\left({x}{,}{y}{,}{z}\right){,}\left({3}{,}{2}{,}{2}\right){=}{0}{,}\left({3}{,}{2}{,}{3}\right){=}{0}{,}\left({3}{,}{3}{,}{1}\right){=}{0}{,}\left({3}{,}{3}{,}{2}\right){=}{0}{,}\left({3}{,}{3}{,}{3}\right){=}{0}\right]\right)\right]\right)$ (2)

Note that the result uses antisymmetric indexing:

 > $\mathrm{op}\left(1,\mathrm{get_compts}\left(\mathrm{ex_der}\right)\right)$
 ${\mathrm{antisymmetric}}$ (3)